The London Sinfonietta with Micachu and the Shapes. Photograph: Rough Trade

In his book The Rest Is Noise, Alex Ross suggests: "One possible destination for 21st-century music is a final 'great fusion': intelligent pop artists and extroverted composers speaking more or less the same language." What he didn't mention was that, since any language needs its interpreters, intelligent, extroverted orchestras might be required to communicate that fusion.

Step up the London Contemporary Orchestra. Its directors – 25-year-old conductor Hugh Brunt and 26-year-old violist Robert Ames – established the outfit in 2008 with a mission to "think very differently about what people want to listen to". They themselves listen to everything from Aphex Twin to Brahms, Foals to Xenakis, and it shows in the range of their work. Over the past year they've curated concerts featuring pieces by Radiohead's Jonny Greenwood and turntable-manipulating young composer Shiva Feshareki alongside Messiaen and John Cage, toured with Belle & Sebastian and recorded a new version of Foals' Spanish Sahara. "The crossover that happens between popular and more formal music is really natural to us," says Ames. "We pinpoint where it happens."

In some ways, the boundaries between the three orchestras are blurry – the LCO and London Sinfonietta share the same pool of musicians; the LCO and the Heritage both perform with pop bands. Yet there are also differences between their approaches to pop-classical fusions: they might all be fluent in the musical language, but their translations vary.

You feel those differences as soon as you meet the people in charge of each orchestra. Brunt and Ames are poised and thoughtful, friendly but restrained. The pair met as teenagers, performing with the National Youth Orchestra. From there, Ames spent six years studying at the Royal Academy of Music while Brunt went to Oxford. If their backgrounds suggest elitism, the mantra of the LCO is accessibility. They seek out "the kind of spaces where you can make an audience feel comfortable", Brunt says: not traditional concert halls but more rough-and-ready venues such as the Village Underground in east London, or the Old Vic Tunnels, beneath Waterloo station. The atmosphere here, says Ames, is more like that of "an event or a gig, so you can clap along, or go and have a drink if you don't want to listen to the piece".

A typical LCO show, says Brunt, is like "a mixtape": its programme for Reverb at the Roundhouse next March, for instance, juggles new work by Greenwood and Prokofiev with a dramatic concerto by Vivier, one of Xenakis's experiments in sound and an electronic breather from Stockhausen in the middle. This suggests a question: when the LCO's musicians are more used to performing the challenging contemporary-classical repertoire, aren't they bored when it comes to playing pop arrangements? "It uses a different set of skills," says Ames, tactfully. At the romantic end of the spectrum, players aim to create "a golden, silky-smooth sound – and doing that is really difficult. While rhythmic music is incredibly hard to get in time." The LCO's leader, violinist Daniel Pioro, is more open. For him, the "technical ease" of playing with Belle & Sebastian, say, is as good as a holiday. "Soloists go through this incredibly rigorous training: sometimes we forget that the ultimate purpose of music is entertainment," he admits. Looking at footage of the LCO dancing along to The Boy With the Arab Strap on stage, you can see why Pioro, a 25-year-old who started playing violin when he was four, says: "That kind of gig makes you younger."

If the LCO holds the formal classical tradition in low regard, the Heritage Orchestra are even less respectful – and its founders, Jules Buckley and Chris Wheeler, are more impish and outspoken to boot. The pair met studying tuba and trombone respectively at the Guildhall School of Music and Drama, but discovered their shared interests lay more in clubbing than the great orchestral heritage – hence the sardonic name. "We liked dance music, we liked rock bands," says Wheeler, so it made sense to experiment with running an orchestra as though it were a gigantic pop group. They had their own club night in London until 2005, and released an album of cinematic, soul-and-disco-inflected compositions by Buckley in 2006.

But what chiefly occupies the orchestra, is its work as a backing band: they've appeared live with a motley selection of acts, including Dizzee Rascal, Antony and the Johnsons, Tim Minchin, and recently had a No 1 hit with Professor Green.

For an outfit with such a strong sense of its own identity, it's hard to see what the appeal is of playing second fiddle, as it were, to a star attraction. What drives Wheeler is a desire to change the way orchestras contribute to pop music. "Generally, orchestral writing in the pop world could be a hell of a lot more progressive," he says. Most of it, says Buckley, comprises what he disparagingly calls "egg notes – long notes in the shape of an egg in the bar". He aims to put more variety into his arrangements, making them less romantic and more rhythmic: "monsterising the groove," as he puts it, "so that you get a group of 30 or 40 people playing like one big fucking drum kit".

Most of this talk is anathema to Andrew Burke, the chief executive of the London Sinfonietta since 2007. "I don't want the Sinfonietta ever to be a backing band," he says emphatically, "making sounds that could have been made on a synth. I would hate to think that we could only collaborate with someone if we just became a larger sonic palette for somebody else's aesthetic." You might get the impression Burke doesn't like pop music, but you'd be wrong: he unexpectedly and somewhat sheepishly confesses to a fondness for Take That and Earth, Wind and Fire. But he also leads an organisation that came into being 40 years before the LCO, has Arts Council funding, and feels a corresponding responsibility not just to deliver new musical languages but help create them.

For Burke, the 2010 collaboration between the Sinfonietta and Micachu was pretty much perfect. Frontwoman Mica Levi brought ideas to the rehearsal room, but that was it. "There wasn't anything written down at the start of the rehearsal process, four days before the gig," marvels Burke. "That's scary – scary in our world." In effect, Levi invited the classical musicians to compose their own contributions. As a result, says Burke: "The integration of the Sinfonietta players was an honest one, in terms of the way the music had been made and in terms of sonic texture."

What all three orchestras are keen to avoid is the potential pitfall of any fusion music: a saggy lack of identity. Burke speaks for all of them when he describes the risk of collaborations: "You can end up with a diluted nonsense that isn't quite contemporary classical and isn't quite experimental pop." What's thrilling about their experiments with new composers, venues and collaborations is that there is no sense of boundary: their vision of music is limitless. "We're particularly fond of compartmentalising the arts in Britain," says Wheeler. "We delineate between visual arts, theatre, music, the subgenres of music. But we [the Heritage] don't really care about genres. Just do what you're doing really well."

The Heritage Orchestra and the Leisure Society are at the Barbican, London EC2 (020-7638 8891), on 8 December. The London Contemporary Orchestra perform at Shoreditch Church, London E1 (020-7377 1362), on 12 December.